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fraseriainlewis / towardsdatascience

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Advantages of bayesian diffusion modelling #1

Open Shuo-HAO opened 3 years ago

Shuo-HAO commented 3 years ago

Dear Fraser Lewis,

It's very glad to see your article for Bayesian diffusion modelling, I'm a PhD student in structural engineering division, more specifically, structural dynamics. I have learned Bayesian based regression method, the Gaussian Process. What I would like to ask is what's the advantages of the Bayesian diffusion modelling compared with Gaussian Process. Wish your reply.

Sincerely

Shuo

fraseriainlewis commented 3 years ago

Hi Shuo, the short answer is flexibility - and I'm no expert on Gaussian processes - but in a Gaussian process all data are assumed jointly Gaussian distributed, i.e. if you have data at 10 times points then these are assumed to be multivariate Gaussian distributed. That is an enormous assumption. The Gaussian distribution is by widely used but this is largely because of historical mathematical convenience (the Gaussian distribution is mathematically tractable and things like the central limit theorem mean that in some circumstances it could be a good assumption). But in general the world is not Gaussian distributed - which is not such a flexible distribution particularly in a multivariate sense as it does not allow for fat tails - that's wishful thinking, but it's convenient as it keeps things as simple as possible. Simple is beautiful. SDEs on the other hand do not make any such assumptions, you define a stochastic process via drift and diffusion terms and the joint distribution of the data could be Gaussian or highly non-Gaussian or entirely non-standard density, it depends on the precise form of the SDE. This makes SDEs much harder to work with, but it's the price for flexibility. I hope this helps! It's also just an option, and I'm sure others might differ :-)

On Fri, 16 Apr 2021 at 14:33, Shuo-HAO @.***> wrote:

Dear Fraser Lewis,

It's very glad to see your article for Bayesian diffusion modelling, I'm a PhD student in structural engineering division, more specifically, structural dynamics. I have learned Bayesian based regression method, the Gaussian Process. What I would like to ask is what's the advantages of the Bayesian diffusion modelling compared with Gaussian Process. Wish your reply.

Sincerely

Shuo

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Shuo-HAO commented 3 years ago

Hi Fraser, Thanks you for your reply. I agree with you, the Gaussian Process is a quite convenient (get rid of integration in marginal likelihood thanks to the mathematical expression of Gaussian distribution) and easy-trained method, it's derived from the Bayesian inference applying on multivariate Gaussian distribution, and I do understand the assumption on the variables as Gaussian can be less rigorous.

I'm very happy to see your article to introduce things from another point of view, SDEs for me is a new thing, and I will consider more specifically for its application on structural dynamic character identification.

Wish you all the best.